Determine whether the cluster-to-x-ferromagnet transition is symmetry-enriched

Determine whether the Ising transition between the symmetry-protected topological cluster phase and the x-direction ferromagnetic phase of the interacting cluster Ising chain is of the symmetry-enriched type.

Background

For positive nearest-neighbor interaction strength, the interacting cluster Ising chain exhibits a transition from the symmetry-protected topological cluster phase to an x-direction ferromagnetic phase. The transition is numerically characterized by Ising criticality with central charge c = 1/2.

Because the protecting Z2 × Z2 symmetry remains unbroken across this transition, the authors note that it may belong to a symmetry-enriched class of Ising transitions. They explicitly leave the determination of whether this characterization applies to the transition for future work.

References

This is an Ising transition between the cluster SPT phase and an Ising ferromagnet, at which the protecting $\mathbb{Z}_2\times\mathbb{Z}_2$ symmetry remains unbroken, as in Ref.. It may, therefore, be of the symmetry-enriched kind, a question we leave for future work.

— Critical bifurcation and deconfined quantum criticality in an interacting cluster Ising chain  (2608.24248 - Sourabh et al., 25 Aug 2026) in Footnote in Section 3, immediately following the discussion of the cluster-to-x-FM transition in Section 3.1